Prove that the sum of the measures of the interior angles of a triangle is 180°. Be sure to create and name the appropriate geometric figures.
Answers
Answer:
Step-by-step explanation:
We can prove it through different facts but we will use the fact of alternate interior angles formed by a transversal with two parallel lines are congruent.
Look at the figure for brief understanding.
Construct a line through B parallel to AC. Angle DBA is equal to CAB because they are a pair of alternate interior angle(alternate interior angles are two interior angles which lie on different parallel lines and on opposite sides of a transversal) The same reasoning goes with the alternate interior angles EBC and ACB....
The sum of the interior angles of a triangle is 180 degrees, which can be demonstrated using properties of parallel lines, alternate interior angles, and Euclidean geometry axioms. By drawing a parallel line and using congruent angles, we show that the sum of angles in a triangle aligns with the angle sum on a straight line.
One classic proof that the sum of the interior angles of a triangle is 180 degrees involves drawing a parallel line to one side of the triangle through the opposite vertex. Let's name the vertices of our triangle A, B, and C. Extend a line from vertex C that is parallel to the line AB. This forms alternate interior angles with the angles at vertices A and B, which we know are equal because of the properties of parallel lines.
Call the angles at A and B in the triangle, angle A and angle B, respectively. Outside of the triangle, we have angles formed between the extended line and the lines AC and BC, let's call these angles A' and B'. By the property of parallel lines and angles, angle A is congruent to angle A' and angle B to angle B'. The line through C forms a straight line, so angle A' plus angle C plus angle B' must equal 180 degrees. Since angle A is congruent to angle A' and angle B to angle B', we can then say that angle A plus angle B plus angle C equals 180 degrees. This is because the sum of the interior angles at A and B and the newly defined angle C is equal to the sum along the straight line, which is always 180 degrees.
Moreover, considering the Euclidean geometry axioms, we know that the sum of angles in a triangle is inherently 180 degrees, and this can be seen in the equilateral triangle example where if we take the large triangle and divide it into four smaller congruent triangles, each of these smaller triangles also has the property that the sum of its angles equals 180 degrees. When we add up the angles from the four small triangles and subtract the sum of the straight angles formed at the large triangle's sides, the result confirms the sum for the large triangle is equivalent to four times the sum for one small one, reinforcing the 180-degree sum rule for each triangle.
Related Questions
Determine whether the given value is a statistic or a parameter. A sample of employees is selected and it is found that 55 % own a vehicle. Choose the correct statement below. a. Parameter because the value is a numerical measurement describing a characteristic of a sample. b. Statistic because the value is a numerical measurement describing a characteristic of a sample. c. Statistic because the value is a numerical measurement describing a characteristic of a population. d. Parameter because the value is a numerical measurement describing a characteristic of a population.
Answers
Answer:
b. Statistic
Step-by-step explanation:
b. Statistic because the value is a numerical measurement describing a characteristic of a sample.
Answer:
Option b
Step-by-step explanation:
Given that a sample of employees is selected and it is found that 55 % own a vehicle.
Before we answer the questions let us understand the difference between a parameter and a statistic.
Parameters are numbers that summarizes the data of a population. But statistics are numbers that summarizes the data of a sample.
Sample is a subset of population i.e. a small portion of the whole population is sample.
Here 55% is the proportion of the sample of employees. Since this is a number summarizing the data about a sample this is called statistic.
b. Statistic because the value is a numerical measurement describing a characteristic of a sample.
(QUICK!!!!!!!!!!!!) Write an equation of the line below.
Answers
Answer:
[tex]\large\boxed{y=\dfrac{3}{5}x-2}[/tex]
Step-by-step explanation:
The slope-intercept form of an equation of aline:
[tex]y=mx+b[/tex]
m - slope
b - y-intercept
The formula of a slope:
[tex]m=\dfraxc{y_2-y_1}{x_2-x_1}[/tex]
From the graph we have the points:
(-5, -5)
y-intercept (0, -2) → b = -2
Calculate the slope:
[tex]m=\dfrac{-2-(-5)}{0-(-5)}=\dfrac{3}{5}[/tex]
Put the value of the slope and the y-intercept to the equation of a line:
[tex]y=\dfrac{3}{5}x-2[/tex]
Chung has 6 trucks and 5 cars in his toy box. Brian has 4 trucks and 5 cars in his toy box. Which is the correct comparison of their ratios of trucks to cars?
Answers
Final answer:
Chung has a ratio of 6 trucks to 5 cars (6:5), and Brian has a ratio of 4 trucks to 5 cars (4:5). Chung has a higher ratio of trucks to cars compared to Brian.
Explanation:
To compare the ratios of trucks to cars for Chung and Brian, we simply write down the number of trucks and cars each has and form a ratio for each. For Chung, the ratio of trucks to cars is 6 trucks to 5 cars, which can be written as 6:5 or ⅓. For Brian, the ratio of trucks to cars is 4 trucks to 5 cars, or 4:5 or ⅔.
Now, by comparing these two ratios, we see that Chung has a higher ratio of trucks to cars (6:5) compared to Brian (4:5), which means Chung has more trucks relative to cars in his toy box than Brian does.
To compare Chung's and Brian's ratios of trucks to cars, we have to calculate each ratio and then compare them.
**Chung's Ratio:**
Chung has 6 trucks and 5 cars. The ratio of trucks to cars for Chung is the number of trucks divided by the number of cars.
Chung's trucks to cars ratio = Number of trucks / Number of cars
= 6 / 5
**Brian's Ratio:**
Brian has 4 trucks and 5 cars. The ratio of trucks to cars for Brian is the number of trucks divided by the number of cars.
Brian's trucks to cars ratio = Number of trucks / Number of cars
= 4 / 5
Both ratios are to be compared now.
**Comparing Ratios:**
Chung's ratio is 6/5, which is 1.2 when converted into a decimal.
Brian's ratio is 4/5, which is 0.8 when converted into a decimal.
Since 1.2 (Chung's ratio) is greater than 0.8 (Brian's ratio), we can conclude that Chung has a higher ratio of trucks to cars compared to Brian.
Therefore, the correct comparison of their ratios is: "Chung has a higher ratio of trucks to cars than Brian."
Determine if the sequence is algebraic or geometric, and find the common difference or ratio.
x 1 2 3 4
f(x) 3 9 27 81
A.) Algebraic, common difference = 3
B.) Algebraic, common difference = 6
C.) Geometric, common ratio = 3
D.) Geometric, common ration = 6
Answers
Answer:
Option C.) Geometric, common ratio = 3
Step-by-step explanation:
we know that
In a Geometric Sequence each term is found by multiplying the previous term by a constant
The constant is called the common ratio
In this problem we have
For x=1, f(1)=3
For x=2, f(2)=9
For x=3, f(3)=27
For x=4, f(4)=81
so
f(2)/f(1)=9/3=3 -----> f(2)=3*f(1)
f(3)/f(2)=27/9=3 -----> f(3)=3*f(2)
f(4)/f(3)=81/27=3 -----> f(4)=3*f(3)
f(n+1)/f(n)=3 -----> f(n+1)=3*f(n)
therefore
This is a Geometric sequence and the common ratio is equal to 3
The sequence is a geometric sequence with a common ratio of 3.
Explanation:The sequence given is a geometric sequence. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In this case, the common ratio can be found by dividing a term in the sequence by its preceding term. For example, if we divide the second term (9) by the first term (3), we get 3. The same goes for the rest of the terms in the sequence. Therefore, the correct answer is Option C: Geometric, common ratio = 3.
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Can someone help me with this math question
Answers
Answer:
2/3
Step-by-step explanation:
These two figures are similar since they have the same shape but not the same size
Yellow figure is larger than the orange figure therefore, the yellow figure is a larger or a dilated version of the orange figure.
Scale factor = Small side
Large side
Scale factor = 10/15
Scale factor = 2/3
The scale factor of this dilation is 2/3. The orange figure is dilated 2/3 times to form the yellow figure.
!!
Answer:
It's 2/3
Step-by-step explanation:
Trust me
What is the sum of the first 8 terms of the geometric series:
3+6+12+24+
0765
382
286
440
Answers
Answer:
765.
Step-by-step explanation:
Sum of n terms = a1 (r^n - 1) / (r - 1) where a1 = the first term and r = the common ratio.
Here r = 6/3 = 2 and a1 = 3.
Sum of 8 terms = 3 * ( 2^8 - 1) / 2 -1)
= 3 * 255
= 765 (answer).
What is the next number in the sequence? 9….3….1….1/3…
Answers
The pattern in the sequence 9, 3, 1, 1/3 involves each number being a third of the previous number. Following this rule, the next number after 1/3, obtained by dividing by 3, is 1/9.
Explanation:To determine the next number in the sequence 9, 3, 1, 1/3, we need to identify the pattern or rule that is being followed. Observing the sequence, each subsequent number appears to be a third of the previous number. The first number is 9, and dividing by 3 gives us 3. Dividing the second number, 3, by 3 gives us 1.
Similarly, dividing the third number, 1, by 3 gives us 1/3. Following this logic, to find the next number in the sequence, we divide 1/3 by 3.
Using the arithmetic of division with fractions, we have (1/3) ÷ 3 = (1/3) ÷ (3/1) = 1/9. Therefore, the next number in the sequence is 1/9. We can assume that the rule being applied in this sequence is to divide each number by 3 to find the next number, which aligns with the mathematical pattern identification techniques commonly used.
The next number in the sequence is 1/9.
To find the next number in the sequence 9, 3, 1, 1/3, we need to identify the pattern. This sequence is a geometric sequence where each term is obtained by multiplying the previous term by a common ratio.
Step-by-step:
Start with 9.Multiply it by the common ratio to get the next term.9× (1/3) = 33 ×(1/3) = 11 ×(1/3) = 1/3To find the next term, we continue this pattern:
1/3× (1/3) = 1/9
Which statement is true?
All rectangles are squares.
All squares are rectangles.
All quadrilaterals are rectangles.
All parallelograms are rectangles.
Answers
Answer:
B. All squares are rectangles.
Step-by-step explanation:
B is the correct answer, because all the squares are rectangles have 4 sides.
The accurate statement is that all squares are rectangles.
The statement that is true among the options provided is All squares are rectangles. This is because squares have all the properties of a rectangle, which is a quadrilateral with four right angles, but with the additional property of having all four sides of equal length. Therefore, because a square fulfills all the criteria of a rectangle, we can conclude that all squares are indeed rectangles. On the other hand, not all rectangles are squares since rectangles do not require all sides to be equal, only that the opposite sides are equal. Similarly, not all quadrilaterals are rectangles because other quadrilaterals, like rhombuses or kites, do not have the necessary four right angles. Finally, while all rectangles are parallelograms (a quadrilateral with opposite sides that are equal and parallel), not all parallelograms have right angles and thus are not all rectangles.
To elaborate, a rectangle is defined as a parallelogram with right angles. When it comes to comparing areas, the area of a rectangle is calculated by multiplying its base by its height. And in the case of squares, since all sides are equal, it's just the side length squared. However, when you have two shapes with equal area -- for instance, a square and a rectangle -- the one with the longer perimeter would be the one with the less compact shape, which in most cases would be the rectangle unless it is also a square.
A company manufactures three types of cabinets. It makes 110 cabinets each week. In the first week, the sum of the number of type-1 cabinets and twice the number of type-2 cabinets produced was 10 more than the number of type-3 cabinets produced. The next week, the number of type-1 cabinets produced was three times more than in the first week, no type-2 cabinets were produced, and the number of type-3 cabinets produced was the same as in the previous week.
In the first week, the number of type-1 cabinets produced was , the number of type-2 cabinets produced was , and the number of type-3 cabinets produced was .
Type cabinet 1 answer - 10 , 15 , 25 ,30
Type 2 - 20 , 25 , 30 , 45
Type 3 - 35 , 45 , 55, 65
WHEN YOU COMMENT PLEASE BE ONE HUNNA PERCENT!! AND JUST GIVE ME THE ANSWER
Answers
Answer:
In the first week, the number of type-1 cabinets produced was 15, the number of type-2 cabinets produced was 30, the number of type-3 cabinets produced was 65.Step-by-step explanation:
If we let a, b, c represent the numbers of type-1, type-2, and type-3 cabinets produced in the first week, respectively, we can write three equations in these unknowns:
a + b + c = 110 . . . . . total cabinets for the first weeka + 2b - c = 10 . . . . relationship of quantities in the first week3a +0b +c = 110 . . . . total cabinets in the second weekIt can be convenient to let a machine solver find the solution to this set of equations. Most graphing calculators can handle it, along with several web sites.
__
Solving by hand, we can subtract the second equation from twice the first. This gives ...
2(a +b +c) -(a +2b -c) - 2(110) -(10)
a +3c = 210 . . . . simplify
Subtracting this from 3 times the third equation gives ...
3(3a +c) -(a +3c) = 3(110) -(210)
8a = 120 . . . . . simplify
a = 15 . . . . . . . divide by 8
Using this in the third equation of the original set, we have ...
3·15 +c = 110
c = 65 . . . . . . subtract 45
Then, in the first equation, we get ...
15 + b + 65 = 110
b = 30 . . . . . . . subtract 80
The solution is (type-1, type-2, type-3) = (15, 30, 65) for the first week.
The problem provides a set of linear equations. Solving this system by substitution or elimination method will give the number of cabinets of each type produced each week. The equations are formed based on the conditions provided in the problem.
Explanation:From the information provided, we can use a system of equations to solve this. Let's denote the number of type-1 cabinets made in the first week as x , the number of type-2 cabinets as y, and the number of type-3 cabinets as z. The first condition in the problem gives us the equation x + 2y = z + 10. The total number of cabinets produced in each week is 110, so we get the equation x + y + z = 110 for the first week. From the second week's conditions, we know that no type-2 cabinets were made (y=0), the number of type-3 cabinets was the same (z=z) and the number of type-1 cabinets was three times as much as the first week (x=3x), this means 3x + 0 + z = 110.. Solving this system of equations will provide the number of cabinets produced for each type.
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Applying the ceiling function. Please help me. 30 points to who can answer me correctly please.
Answers
Explanation:
A fraction of an hour costs the same as an hour.
actual time ⇒ time charged ⇒ cost of parking
5 min ⇒ 1 hour ⇒ $3
1 hour ⇒ 1 hour ⇒ $3
1 hour 50 min ⇒ 2 hours ⇒ $6
2 hours ⇒ 2 hours ⇒ $6
2 hours 1 min ⇒ 3 hours ⇒ $9
3 1/2 hours ⇒ 4 hours ⇒ $12
Xanthia buys hot dogs that come in packages of six, and she buys hot dog buns that come in packages of eight. What is the smallest number of hot dog packages she can buy in order to be able to buy an equal number of hot dogs and hot dog buns?
Answers
Answer:
4.
Step-by-step explanation:
The smallest number of hot dogs packages and hot dog buns that has the same amount of is the least common multiple between 6 and 8.
6 = 3*2
8 = [tex]2^{3}[/tex]
So, the least common multiple is the product of each multiple with the biggest exponent, that is [tex]2^{3}*3=24[/tex]. Then, Xanthia has to buy 4 hot dog packages to have 24 hotdogs and 24 hotdog buns.
To solve this problem, we first find the least common multiple (LCM) of 6 and 8. 6=2*3 and 8=2^3, so their LCM is 2^3*3=24. Therefore, Xanthia can buy 24÷6=4 hot dog packages and 24÷8=3 hot dog bun packages to have an equal number of hot dogs and hot dog buns. So you have an answer of 4.
Please give me brainliest, I'm trying to get to a higher rank... You don't have to tho, I respect that someone else deserves it sometimes.
What is the explicit rule for the geometric sequence? 4.05, 1.35, 0.45, 0.15, ...
an=4.05(13)n−1
an=4.05(13)n+2
an=4.05(13)n
an=4.05(13)n+1
Answers
Answer: First option is correct.
Step-by-step explanation:
Since we have given that
4.05, 1.35, 0.45, 0.15, ...
Since it is a geometric sequence.
So, here, a = 4.05
r = [tex]\dfrac{a_2}{a_1}=\dfrac{1.35}{4.05}=0.33[/tex]
So, we know the formula for nth term in geometric sequence.
[tex]a_n=ar^{n-1}\\\\a_n=4.05(0.3)^{n-1}[/tex]
Hence, First option is correct.
Answer:
an=4.05^(1/3)n−1
Step-by-step explanation:
The Discriminant of a quadratic equation is -6. What types of solutions does the equation have?
Answers
Answer:
2 complex conjugates
Step-by-step explanation:
The discriminate is the part of the quadratic formula that is under the radical sign. If the discriminate is negative, that means that the solutions, both of them, are complex conjugates, aka imaginary solutions.
For this case we have that by definition, the discriminant of an equation is given by:
[tex]D = b ^ 2-4 (a) (c)[/tex]
We have the following cases:
[tex]D> 0:[/tex] Two different real roots
[tex]D = 0:[/tex]Two equal real roots
[tex]D <0:[/tex] Two different complex roots
In this case we have to:
[tex]D = -6[/tex], [tex]-6 <0[/tex] , Then we have two different complex roots.
Answer:
OPTION B
There are 81 cars in the CMC parking lot, which are all Acuras, Beetles, or Camrys. There are half as many Acuras as Beetles. The number of Camrys is 80\% of the number of Acuras and Beetles together. How many of the 81 cars are Beetles?
Answers
Answer: 30
Step-by-step explanation:
Let x be the number of Beetles.
Then , the number of Acuras = [tex]\dfrac{1}{2}x[/tex]
Also, The number of Camrys is 80% of the number of Acuras and Beetles together.
Thus , the number of Camrys =[tex]0.8(x+\dfrac{1}{2}x)[/tex]
Now, the total number of cars in parking lot will be :-
[tex]x+\dfrac{1}{2}x+0.8(x+\dfrac{1}{2}x)=81\\\\\Rightarrow\ \dfrac{3x}{2}+0.8(\dfrac{3x}{2})=81\\\\\Rightarrow\ \dfrac{3x+2.4x}{2}=81\\\\\Rightarrow\ 5.4x=2\times81\\\\\Rightarrow\ x=\dfrac{162}{5.4}=30[/tex]
Hence, there are 30 Beetles.
Answer:
30 of the cars
Step-by-step explanation:
I just did the question on Alcumus.
Hope this helped! :)
please help, sorry for it being hard to read.. 1st correct answer gets branliest
Answers
Answer:
86 because 180 subtract 94 is 86
Answer:
The correct answer is option D. 94 °
Step-by-step explanation:
From the figure we get,
l║ m ║ o and n║ p
To find the measure of <16
It is given that, m<1 = 94°
m<1 + m<4 = 180 [ Same side exterior angles of parallel lines are supplementary]
m<4 = 180 - m<1 = 180 - 94 = 86°
Similarly, m<4 + m<16 = 180
m<16 = 180 - m<4 = 180 - 86 = 94°
The correct answer is option D. 94 °
For 20points.
============
A. 25°
B. 30°
C. 35°
D. 40°
Answers
Answer:
A. 25°
Step-by-step explanation:
The angles on either side of the bisector are congruent, so ...
(3x -5)° = (x +15)°
2x = 20 . . . . . . . . . . . . divide by °; add 5-x
x = 10 . . . . . . . . . . . . . .divide by 2
Substitute this result into the expression for the angle measure:
m∠BAC = (3·10 -5)° = 25°
A line passes through the point (-4,3) and has a slope of -4. Write an equation in slope-intercept form for this line. ( Please help!!!!!)
Answers
[tex]\bf (\stackrel{x_1}{-4}~,~\stackrel{y_1}{3})~\hspace{10em} slope = m\implies -4 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-3=-4[x-(-4)]\implies y-3=-4(x+4)[/tex]
[tex]\bf y-3=-4x-16\implies y=-4x-13\impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}[/tex]
A sample of 4 different calculators is randomly selected from a group containing 17 that are defective and 37that have no defects. What is the probability that at least one of the calculators is defective?
Answers
Answer: 0.8025
Step-by-step explanation:
Given : The number of defective calculators : 17
The number of calculators are not defective : 37
Total calculators : 37+17=54
The probability of the calculators are defective : [tex]\dfrac{17}{54}=\dfrac{1}{3}[/tex]
Binomial distribution formula :-
[tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex], where P(x) is the probability of success in x trials, n is total trials and p is the probability of success for one trial.
The probability that at least one of the calculators is defective is given by :-
[tex]P(x\geq1)=1-P(0)\\\=1-(^4C_0(\dfrac{1}{3})^0(1-\dfrac{1}{3})^4)\\\\=1-(\dfrac{2}{3})^4=0.80246913\approx0.8025[/tex]
Round to estimate the answer, and then solve to find the correct answer. Explain whether your estimate was reasonable: A T. Rex dinosaur eats ten twelfths of a plant and then eats two twelfths of the plant later. How much of the plant did the dinosaur eat in total?
Answers
Answer:
1 plant
Step-by-step explanation:
The T.Rex dinosaur eats ten twelffths of a plant and later eats two twelfths of a plant. In total, the T.Rex dinosaur ate:
[tex]\frac{10}{12} + \frac{2}{12} =\frac{12}{12}=1[/tex]
Therefore, the dinsaur in total ate one entire plant.
It's better to solve the problem by using fractions instead of decimals. If we had used decimals the response would be the following:
[tex]0.833333+0.166666=0.999999[/tex] which can be rounded to 1.
This is just a square, help
Answers
Answer:
Step-by-step explanation:
They are all true
the answer is a
a square should have 4 lines or sides the same length any longer or shorter would make it a rectangle
the difference in the x-coorinates of two points in 3 and the difference in the y-coorinates of two points is 6 what is the slope of the line that passes through the points
Answers
Answer:
The slope is 2
Step-by-step explanation:
The slope of a line passing through two points is given by:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
which is the difference between the y-coordinates of two points divided by the difference between x-coordinates of two points
We are given:
Difference between x-coordinates = 3
Difference between y-coordinates = 6
So,
[tex]m = \frac{6}{3}\\=2[/tex]
The slope of the line that passes through these points is 2 ..
What is the slope of the line in this graph? a.5/9 b.5/7 c. 7/5 d.9/7
Answers
Answer:
b. 5/7
Step-by-step explanation:
The line goes through the points (0, 0) and (7, 5). Let's use those points in the slope formula:
[tex]m=\frac{5-0}{7-0}=\frac{5}{7}[/tex]
The slope of that line is 5/7
The slope of the given line is 5/7
What is slope of a line?Slope of a line is the inclination of that line towards the x-axis.How to find the slope of the given line ?We know that slope of a line passing through the points (a, b) and (c ,d), can be given by the formula: [tex]\frac{d-b}{c-a}[/tex]In the given graph, the line passes through the center (0, 0) and (7, 5)
So the slope will be [tex]\frac{5-0}{7-0} = \frac{5}{7}[/tex]
So option B is correct
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What is the approximate length of arc s on the circle below? Use 3.14 for pi. Round your answer to the nearest tenth.
A. 5.6 in.
B. 6.3 in.
C. 14.3 in.
D. 25.1 in.
Answers
Answer:
B 6.3
Step-by-step explanation:
r = 8
l = 2*3.14*r
l = 50.24
s=l/360*45
s≈6.3
For this case we have that by definition, the arc length is given by:
[tex]Al = 2 \pi * r *\frac {a} {360}[/tex]
Where:
r: It's the radio
a: It is the angle of the sector
Then, according to the data we have:
[tex]Al = 2 \pi * 8 * \frac {45} {360}\\Al = 2 * 3.14 * 8 * \frac {45} {360}\\Al = 50.24 * \frac {45} {360}\\Al = 6.28[/tex]
Rounding we have 6.3in
Answer:
Option B
Find the equation in slope-intercept form that describes a line through (–1, 1) and (2, 3)
Answers
Answer:
y = 2/3x + 5/3
Step-by-step explanation:
The slope of the line is ...
slope = (change in y)/(change in x) = (3-1)/(2-(-1)) = 2/3
Then the point-slope form of the desired line can be written ...
y = m(x -h) +k . . . . . slope m through point (h, k)
y = 2/3(x +1) +1 . . . . slope 2/3 through point (-1, 1)
y = 2/3x + 5/3 . . . . . . simplify to slope-intercept form
The equation that describes a line through points (-1, 1) and (2, 3) in slope-intercept form is y = 2/3x + 5/3, determined by calculating the slope and y-intercept.
Explanation:The question asks to find the equation in slope-intercept form that describes a line through (-1, 1) and (2, 3). In order to do this, we need to find the slope and y-intercept of the line.
The slope of the line (m) can be determined by using the formula m = [tex](y_2 - y_1) / (x_2 - x_1)[/tex]. Inserting the given points into this formula gives: m = (3 - 1) / (2 - (-1)) = 2 / 3 = 2/3.
To find the y-intercept (b), we can use the point-slope form of the equation and solve for 'b', y = mx + b, insert the slope we found and one of the given points, let's utilise (-1, 1): 1 = 2/3*(-1) + b, which simplifies to b = 5/3.
So, the equation of the straight line in slope-intercept form is y = 2/3x + 5/3.
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Find the total area of the solid figure.
Answers:
90 sq. ft.
126 sq. ft.
150 sq. ft.
Answers
Answer:
90 sq. ft.
Step-by-step explanation:
To find the "volume," you will need to multiply every the LxWxH.
Length = L
Width = W
Height = H
Then the answer is 90 sq. ft.
Answer:
Surface area = 126 square ft .
Step-by-step explanation:
Given : Rectangular cuboid .
To find : Find the total area of the solid figure.
Solution : We have given Rectangular cuboid .
Length = 3 ft .
Width = 5 ft .
Height = 6 ft .
Surface area = 2 ( l*w + w*h + l *h).
Plug the values
Surface area = 2 ( 3*5 + 5*6 + 3 *6).
Surface area = 2 (15 + 30 + 18).
Surface area = 2 ( 63).
Surface area = 126 square ft .
Therefore, Surface area = 126 square ft .
I need your help with this problem
Answers
Answer:
12.9 m
Step-by-step explanation:
Let d represent the length of the diagonal. Then d-2 is the length and d-6 is the width. The Pythagorean theorem can be used to relate these measures, which are the legs and hypotenuse of a right triangle.
d² = (d-2)² + (d-6)²
d² = d² -4d +4 + d² -12d +36 . . . . eliminate parentheses
0 = d² -16d +40 . . . . . . . . . . . . . . . subtract d², collect terms
0 = d² -16d +64 -24 . . . . . . . . . . . rearrange the constant to make a square
0 = (d -8)² -24 . . . . . . write in vertex form
d -8 = √24 . . . . . . . . . add 24 and take the square root
d = 8 + √24 . . . . . . . . the negative square root is extraneous in this problem
d ≈ 12.9 . . . meters
The length of the diagonal is about 12.9 meters.
please help
Solve for the indicated variable in the literal equation
Ax + By = C for x
Answers
Answer:
x = (C-By)/A
Step-by-step explanation:
Ax + By = C
Subtract By from each side
Ax + By-By = C-By
Ax = C -By
Divide each side by A
Ax/A = (C-By)/A
x = (C-By)/A
Solve, then check algebraically and graphically. 9x-3=78
Answers
Answer:
x=9
Step-by-step explanation:
I have answered ur question
When a certain tree was first planted, it was 4 feet tall, and the height of the tree increased by a constant amount each year for the next 6 years. At the end of the 6th year, the tree was 1/5 taller than it was at the end of the 4th year. By how many feet did the height of the tree increase each year?
Answers
Answer:
[tex]\frac{2}{3}\text{ feet}[/tex]
Step-by-step explanation:
Let the equation that models the height of the tree after x years,
y = mx + c
Where, m is constant amount of increasing and c is any constant,
Given,
When x = 0, y = 4,
⇒ 4 = m(0) + c ⇒ c = 4,
Now, the height of plant after 4th year = m(4) + c = 4m + c
Also, the height of plant after 6th year = m(6) + c = 6m + c
According to the question,
6m + c is [tex]\frac{1}{5}[/tex] more than 4m + c,
[tex]6m+c=4m+c + \frac{1}{5}(4m+c)[/tex]
[tex]6m+c = \frac{6}{5}(4m+c)[/tex]
[tex]30m+5c=24m+6c[/tex]
[tex]6m=c[/tex]
By substituting the value of c
6m = 4
⇒ [tex]m=\frac{4}{6}=\frac{2}{3}[/tex]
Hence, 2/3 feet of height is increased each year.
Maggie graphed the image of a 90 counterclockwise rotation about vertex A of . Coordinates B and C of are (2, 6) and (4, 3) and coordinates B’ and C’ of it’s image are (–2, 2) and (1, 4). What is the coordinate of vertex A. (EXPLAIN WORK)
Answers
Answer:
A(2,2)
Step-by-step explanation:
Let the vertex A has coordinates [tex](x_A,y_A)[/tex]
Vectors AB and AB' are perpendicular, then
[tex]\overrightarrow {AB}=(2-x_A,6-y_A)\\ \\\overrightarrow {AB'}=(-2-x_A,2-y_A)\\ \\\overrightarrow {AB}\perp\overrightarrow {AB'}\Rightarrow \overrightarrow {AB}\cdot \overrightarrow {AB'}=0\Rightarrow (2-x_A)(-2-x_A)+(6-y_A)(2-y_A)=0[/tex]
Vectors AC and AC' are perpendicular, then
[tex]\overrightarrow {AC}=(4-x_A,3-y_A)\\ \\\overrightarrow {AC'}=(1-x_A,4-y_A)\\ \\\overrightarrow {AC}\perp\overrightarrow {AC'}\Rightarrow \overrightarrow {AC}\cdot \overrightarrow {AC'}=0\Rightarrow (4-x_A)(1-x_A)+(3-y_A)(4-y_A)=0[/tex]
Now, solve the system of two equations:
[tex]\left\{\begin{array}{l}(2-x_A)(-2-x_A)+(6-y_A)(2-y_A)=0\\ \\(4-x_A)(1-x_A)+(3-y_A)(4-y_A)=0\end{array}\right.\\ \\\left\{\begin{array}{l}-4-2x_A+2x_A+x_A^2+12-6y_A-2y_A+y^2_A=0\\ \\4-4x_A-x_A+x_A^2+12-3y_A-4y_A+y_A^2=0\end{array}\right.\\ \\\left\{\begin{array}{l}x_A^2+y_A^2-8y_A+8=0\\ \\x_A^2+y_A^2-5x_A-7y_A+16=0\end{array}\right.[/tex]
Subtract these two equations:
[tex]5x_A-y_A-8=0\Rightarrow y_A=5x_A-8[/tex]
Substitute it into the first equation:
[tex]x_A^2+(5x_A-8)^2-8(5x_A-8)+8=0\\ \\x_A^2+25x_A^2-80x_A+64-40x_A+64+8=0\\ \\26x_A^2-120x_A+136=0\\ \\13x_A^2-60x_A+68=0\\ \\D=(-60)^2-4\cdot 13\cdot 68=3600-3536=64\\ \\x_{A_{1,2}}=\dfrac{60\pm8}{2\cdot 13}=\dfrac{34}{13},2[/tex]
Then
[tex]y_{A_{1,2}}=5\cdot \dfrac{34}{13}-8 \text{ or } 5\cdot 2-8\\ \\=\dfrac{66}{13}\text{ or } 2[/tex]
Rotation by 90° counterclockwise about A(2,2) gives image points B' and C' (see attached diagram)